Except as the distance function of complexity geometry, another motivation to define the SDOS is that operator size itself is also an important quantity in the study of quantum chaos [1]}, [2]}, [3]} and quantum gravity [4]}, [5]}, [6]}, [7]}. Especially, it was conjectured that operator size has a gravitational dual which is related to radial momentum [5]}. However, there is an ambiguity that operators of different size can correspond to the same particle in the bulk due to the quantum error correction property of AdS/CFT [9]}, so which operator size is dual to the "momentum"? To answer the question, we should notice that these operators have the same action on the state. The operator size measures the number of local d.o.f. that the operator can change, but doesn't measure how many local d.o.f. are really changed. The latter is measured by SDOS. So although different operator size are related to the same radial momentum, these operators have the same state-dependent operator size. Then the ambiguity can be fixed. So it will be interesting to further study the relation between the SDOS and momentum.
